In a large saucepan or small stockpot over medium heat, warm the oil. Add the onion and cook, stirring occasionally, until starting to become translucent, about 5 minutes.
Add the carrots, celery, garlic, thyme, and celery salt and cook, stirring occasionally, until the carrots and celery are crisp-tender, about 3 minutes.
Add the ham, broth, cabbage, and bay leaves and bring to a boil over high heat.
Reduce to a simmer, cover, and cook until the cabbage is tender, 15 to 20 minutes.
Add salt and pepper to taste, and parsley.
Remove the bay leaves before serving.
*To make ham broth, combine 2 smoked ham hocks, 1 carrot, 1 celery stalk, 1 onion (quartered), 2 bay leaves, 1 teaspoon whole peppercorns and 4 quarts water in a large stockpot. Bring to a boil over low heat, reduce to a very gentle simmer, cover and cook until the meat falls off the ham hocks, 3 to 4 hours. Strain the broth and add salt and pepper to taste. Makes about 3 quarts.
